I think the Robertson is hard to turn down. In addition to having the opportunity to attend a top 10 private university and top 5 public, the social opportunities are immense.
Here’s a comment from last year by a Robertson at UNC I like a lot (although it is not the same as your predicament - the scholar raises a lot of good points) regarding a member who was deciding between Princeton and the Robertson.
http://talk.collegeconfidential.com/princeton-university/1594258-princeton-vs-robertson-scholarship-at-duke-unc.html
Congratulations on all these great opportunities! Before I give my comments, I do want to give my specific background. I am currently a Robertson Scholar at UNC. My choice was between Dartmouth and the Robertson. Given that Dartmouth gave me no money, the Robertson was the obvious choice. I’m very close friends with Duke Robertson Scholars and Princeton students. I would largely agree with all of your disadvantages and advantages that you point out about the Robertson program. I’ll try to recognize my own bias and do my best to recommend accordingly. Also, I do not know much about Cambridge, so I will just not open that can of worms.
I do want to caution on a few things. First, consulting and investment banking are very common for both Princeton graduates and Robertson Scholars. That is less a symptom of each program, but rather the incentive graduates have to make a great deal of money. The Big Three management consulting firms recruit at all three institutions. Second, the Robertson is a young scholarship program. It’s name is well-recognized, but the amount of graduate merit scholars it has produced is low. Why? Because the number of Robertson scholars that have graduated is much lower than the number of Duke-AB or Princeton graduates. So judging the accomplishments of alumni is a hard gauge to judging these institutions. Third, as another comment articulated, these programs are not destinations. In your specific situation, I would be a firm believer that it’s all about what you do, not where you go.
I want to heavily emphasize that the biggest weighing factor is where you would be most happy. I highly recommend visiting these campuses before making your decision. See which you enjoy as far as the environment, people, and discussions are concerned. Any factors of that nature are more important than I can describe. I know this is an unsatisfying answer to many prospective college first-years. When I was in your shoes, I would have been unsatisfied with this response. Furthering a start to success with a better name, better connections, or more recognition is important, but alas, a wash when it comes to these great opportunities. I had friends turn down Harvard for Duke. I had friends turn down Duke for Harvard. In the end, the difference between Princeton and the Robertson comes in one dimension: which makes you so crazy you never want to leave?
These days, graduate school is the name that matters so much more. Undergraduate is important in getting into a good graduate school, but again, this centers on what you do, rather than where you go. Both institutions will give you great opportunities to do what you want to do. There is no doubt. The difference is what environment of greater contentment will foster these accomplishments. Talking to alumni or visiting in-person are the best ways to do this. Good luck and let me know if you have any questions about the Robertson that Google cannot solve. I can even put you in touch with other Brits that are currently Robertsons and that faced similar decisions. Or you may have decided already. Either way, best wishes.